The Platonic Solids and Finite Rotation Groups

SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET The Platonic solids and finite rotation groups av Hanna Rislund 2019 - No K35 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM The Platonic solids and finite rotation groups Hanna Rislund Självständigt arbete i matematik 15 högskolepoäng, grundnivå Handledare: Gregory Arone 2019 Abstract The Platonic solids have fascinated humanity for more than 2000 years. This thesis explores polygons and polyhedra in order to find the regular polyhedra. It turns out there are five of them; the regular tetrahedron, the cube, the regular octahedron, the regular icosahedron and the regular dodecahedron. They are together called the Platonic solids, named after Plato, who wrote about them in his dialogue ”Timaeus”. The thesis also examines the rotation groups of the Platonic solids, as well as the other two finite subgroups of the rotation group SO(3). Acknowledgements I would like to thank my supervisor Gregory Arone, for his help with the theory and for the many examples he gave to help me understand. I also like to thank my referee Jörgen Backelin, for his comments and suggestions on how to improve my thesis. 2 Contents List of Figures 4 1 Introduction 5 1.1 The history of the Platonic solids . .5 1.1.1 Plato, Theaetetus and Euclid . .5 1.1.2 Kepler and Euler . .5 2 Polytopes 7 2.1 Polygons . .8 2.2 Polyhedra . .9 2.3 Schläflisymbol . 13 2.4 Graphs . 13 2.5 Dual polyhedra . 14 2.6 Euler’s formula . 15 3 The Platonic solids 17 3.1 Tetrahedron . 17 3.2 Cube . 18 3.3 Octahedron . 19 3.4 Icosahedron . 19 3.5 Dodecahedron . 24 3.6 The vertices of regular polyhedra . 25 3.7 Five regular polyhedra . 27 4 Rotation groups 29 4.1 Rotations of the tetrahedron . 30 4.2 Rotations of the cube and octahedron . 31 4.3 Rotations of the icosahedron and dodecahedron . 32 4.4 Permutation groups . 33 4.4.1 The tetrahedral group . 34 4.4.2 The octahedral group . 34 4.4.3 The icosahedral group . 34 4.5 Finite subgroups of the rotation group SO(3) . 35 5 Summary 37 6 References of books 38 7 References of figures 39 3 List of Figures 1 Kepler attempted to correlate the Platonic solids to the orbits of the six known planets (at the time). [12] . .6 2 Two hexagons, one convex and one not convex. .8 3 Different polygons. [7] [8] . .9 4 The faces, edges and vertices of a cube. [9] . .9 5 A complete flag of a cube (a vertex, an edge and a face). 10 6 The steps for creating a dual graph. [10] . 14 7 The octahedron is the dual of the cube. [11] . 15 8 By removing edges, the multigraph can be reduced. [12] . 16 9 The five Platonic solids. [12] . 17 10 The dual of a tetrahedron is another tetrahedron. [13] . 18 11 The dual of the cube is the octahedron. [14] . 19 12 By colouring the octahedron alternately black and white like in (a), the icosahedron can be found by dividing the edges of the octahedron according to the golden ratio, like in (b). 20 13 A vertex figure of a cube. [17] . 21 14 An excellent vertex figure of a cube. [18] . 22 15 The icosahedron and the dodecahedron are duals. 24 16 The dodecahedron can be seen as two bowls that are put together. 25 17 A vertex can be created with three, four or five equilateral triangles, but not with six, since six equilateral triangles create a flat construction. [23][24] . 25 18 Five possible vertices of the Platonic solids, laid flat at the top and folded into a vertex at the bottom. [12] . 26 19 The rotation axes through a tetrahedron. [27] . 30 20 The rotation axes through an octahedron. [28] . 31 21 The axes of rotation in the cube and octahedron correspond to each other. [29] . 32 22 The axes of rotation through an icosahedron and a dodecahedron are the same. 33 23 The five edges connected to the same vertex in the icosahedron will belong to five different classes. [30] . 34 4 1 Introduction 1.1 The history of the Platonic solids 1.1.1 Plato, Theaetetus and Euclid Our story of the regular polyhedra will begin around 360 BC. At that time, Plato (429-347 BCE)[3] wrote about them in his dialogue “Timaeus”, and consequently they are called Platonic solids [2]. In the dialogue he linked the regular solids to the elements fire, water, earth and air. The tetrahedron was fire, since it is the sharpest of the solids, it is also the smallest and therefore the driest. The cube was earth, because it is the most stable. The octahedron was air, since it can easily be spun between two fingers and is therefore the most unstable. The icosahedron was water, because of its many sides it flows easily. At last the dodecahedron was the universe, since it is the biggest of the solids and can enclose all the others. The story continues with Theaetetus (417-369 BCE)[3], who was active at Plato’s Academy in Athens. He studied the Platonic solids mathemati- cally and realised that there are only five regular solids. He also gave a proof of this fact, which is probably the first proof of it. The earliest preserved mathematical script that deals with the regular solids is called Elements and is written by Euclid (lived around 300 BCE) [3]. Elements consists of thirteen books, and in the last one Euclid constructs the regular solids and shows how they can be enclosed in a sphere. He also compares the edges of the solids to the radius of the sphere. The proof by Theaetetus, that there are only five regular solids, is included in the book. 1.1.2 Kepler and Euler Throughout the history many scientists have investigated the Platonic solids. One that took a special interest in them was Kepler (1571-1630)[3]. He thought that it must be special that there are only five of them and tried to connect that fact to the rest of the world. In that time only six planets in our solar system were known. These six planets had to be linked to the regular solids according to Kepler. First, he tried to correlate the orbits of the planets to polygons, but soon realized that it did not work. Then he made spheres of the orbits and placed the regular polyhedra inside the spheres. The first sphere was the orbit of Saturn, inside that sphere he inscribed a cube. Inside the cube, he inscribed another sphere that represented the orbit of Jupiter. He then continued with a regular tetrahedron, a sphere, a regular dodecahedron, a sphere, a regular icosahedron, a sphere, a regular octahedron and a sphere. In this way he argued that all the orbits of the planets were represented by the spheres separated by the regular solids. Since he knew that the orbits were not circular, he gave the spheres a thickness to fit better to the data presented 5 by Copernicus (1473-1543). But even then, some parts of his model did not quite fit with the data. He solved this by simply stating that the data was wrong and that his model was right. He published the model in his first book ”Mysterium Cosmographicum” in 1596. An illustration of the model from the book can be seen in Figure 1. Later in his life, Kepler used the same data to discover the laws of planetary motions. The next big step for the Platonic solids where made by Euler (1707- 1783)[3]. He wanted to classify the polyhedra by counting their features. He began with naming the different parts of the solids; the 0-dimensional components he called vertices, the 1-dimensional components he called edges and the 2-dimensional components he called faces. By counting the different components, he found the simple formula V E + F = 2. To his own − knowledge, he was the first to notice this relationship. It is surprising that it took this long for someone to notice this relationship, even though count- less mathematicians have studied the Platonic solids for over 2000 years. But before Euler, the study of the polyhedra was focused on the properties that could be measured; the length, area, volume and angles. No one had explicitly referred to the edges before or tried to classify the polyhedra by the number of vertices, edges and face, and thus no one had counted them in order to compared them. Figure 1: Kepler attempted to correlate the Platonic solids to the orbits of the six known planets (at the time). [12] 6 2 Polytopes In order to understand what the Platonic solids are, this section will present some basic definitions about polygons and polyhedra. We start with some definitions about sets. Definition 2.1. A set is a collection of selected objects. An object x in a set A is called an element of A, written x A. ∈ Definition 2.2. Let A and B be two sets. If every element of A is also an element of B, then A is a subset of B, written A B. ⊆ Definition 2.3. A subset S in Rn is convex if for every two pointsx ¯ andy ¯ in S (1 t)¯x + ty¯ S, for all 0 t 1. − ∈ ≤ ≤ Theorem 2.1. Given any collection of convex sets, their intersection is itself a convex set. Proof. The intersection can either be empty, consist of a single point or consist of more than a single point.

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